A361953 Triangle read by rows: T(n,k) is the number of unlabeled weakly graded (ranked) posets with n elements and rank k.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 6, 1, 0, 1, 20, 30, 9, 1, 0, 1, 55, 145, 66, 12, 1, 0, 1, 163, 745, 465, 111, 15, 1, 0, 1, 556, 4245, 3444, 964, 165, 18, 1, 0, 1, 2222, 27880, 28024, 8618, 1652, 228, 21, 1, 0, 1, 10765, 218058, 259974, 83322, 16569, 2556, 300, 24, 1

Offset: 0

Author

Andrew Howroyd, Mar 31 2023

Keywords

• nonn
• tabl

Comments

Here weakly graded means that there exists a rank function rk from the poset to the integers such that whenever v covers w in the poset, we have rk(v) = rk(w) + 1.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   3,    1;
  0, 1,   8,    6,    1;
  0, 1,  20,   30,    9,   1;
  0, 1,  55,  145,   66,  12,   1;
  0, 1, 163,  745,  465, 111,  15,  1;
  0, 1, 556, 4245, 3444, 964, 165, 18, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..860 (rows 0..40)
• Andrew Howroyd, PARI Program, Apr 2023.
• Wikipedia, Graded poset.

Crossrefs

Row sums are A361920.

The labeled version is A361951.

Cf. A263859, A361952, A361954 (connected).

Programs

PARI

\\ See link for program code.
{ my(A=A361953tabl(8)); for(i=1, #A, print(A[i, 1..i])) }

Formula

G.f. of column k >= 2: C(k,x)/C(k-1,x) - C(k-1,x)/C(k-2,x) where C(k,x) is the g.f. of column k of A361952.